Description of anisotropically microstrain-broadened line profiles by Edgeworth series

Abstract

The diffraction effects within the Stokes-Wilson approximation of a non-Gaussian microstrain distribution in a macroscopically isotropic specimen are described by anisotropic Edgeworth series. This Edgeworth series can for each reflection hkl be quantified by the 3^rd, 4^th ... N^th cumulants of the microstrain distribution projected on the reflection´s diffraction vector. The diffraction-vector (hkl) dependence of these N^th cumulants can be described by 2N^th-rank tensors which characterise the microstrain distribution in the crystal´s frame of reference. The situation considerably simplifies in the special case of univariateness of the microstrain distribution, leading to equal (or inverted) shapes of the line-broadening contribution due to microstrain for all different hkl reflections. The Edgeworth description of microstrain was applied to X-ray powder diffraction data of two slightly inhomogeneous hexagonal ε-iron-nitride powders, ε-FeN_0.433 and ε-FeN_0.407. The microstrain-like line-broadening data was evaluated in terms of the probability-density functions described by Edgeworth series for the composition of these two powders.